Chapters

What Is Interest and How Is It Calculated?
Formulas for Calculating Compound Interest
Proposed Interest Calculation Exercises

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What Is Interest and How Is It Calculated?

In some banks, savings accounts or even in loans and pawns we can observe that a monthly interest is charged. This does not mean that all interests must be like this.

The period is the time in which interest is applied, this can be annual, monthly, weekly, or even daily.

For the case of the exercises proposed in this section, we will be working with annual interest.

Also, there is simple interest and compound interest. The exercises we will see deal with compound interest.

Formulas for Calculating Compound Interest

For the following formulas we have the following notation:

$\text{[math]}$ : Interest
$\text{[math]}$ : Initial capital
$\text{[math]}$ : Interest rate
$\text{[math]}$ : Nominal interest rate
$\text{[math]}$ : Amount (or future value)
$\text{[math]}$ : Time of loan or investment
$\text{[math]}$ : Period of loan or investment
$\text{[math]}$ : Capitalization frequency

Thus, the formulas related to compound interest calculation, when the interest rate and time use the same time unit, are:

$\text{[math]}$
$\text{[math]}$
$\text{[math]}$
$\text{[math]}$
$\text{[math]}$
$\text{[math]}$
$\text{[math]}$

Note that if time and interest use different units, then we will have to make the appropriate conversions before using the formulas.

Proposed Interest Calculation Exercises

Score:

For how long must a capital of $25,000 be invested at 5% quarterly so that with compound interest it becomes $30,387.66?

Solution

Note that we have the amount or final capital, the initial capital and the interest rate

$\text{[math]}$
$\text{[math]}$
$\text{[math]}$ quarterly

We directly apply the formula to calculate time, this will be in quarters since the interest rate is quarterly:

$\text{[math]}$

Therefore, the required time is 4 quarters.

$45,000 is lent with compound interest and after 2 years $52,488 is received. Calculate the interest rate.

Solution

Note that we have the capital, the amount and the investment period

$\text{[math]}$
$\text{[math]}$
$\text{[math]}$ years

We directly apply the formula to calculate the interest rate

$\text{[math]}$

Thus, the interest rate is 8% annually.

Find the compound interest rate (as a percentage) at which a capital should be lent so that after 20 years the interest is equivalent to the lent capital.

Solution

We want the interest to equal the capital, so the amount equals twice the capital

$\text{[math]}$

Note that we have amount, capital and the investment period

$\text{[math]}$
$\text{[math]}$
$\text{[math]}$

Since the capital is different from 0, we cancel $\text{[math]}$ from both sides of the interest rate equation and substitute the investment period:

$\text{[math]}$

Thus, the interest rate is 3.53% annually.

In how much time will the compound interest equal triple the initial capital placed at an interest rate of 6%?

Solution

We want the interest to be triple the capital, so the amount equals four times the capital

$\text{[math]}$

Note that we have amount, capital and interest rate

$\text{[math]}$
$\text{[math]}$
$\text{[math]}$

Since the capital is different from 0, we cancel $\text{[math]}$ in the equation to calculate time

$\text{[math]}$

Therefore, the required time is 23.79 years.

Find the compound interest produced during five years, by a capital of $30,000, at 6%.

Solution

Note that we have the investment period, capital and interest rate

$\text{[math]}$
$\text{[math]}$
$\text{[math]}$

Here we need to find the value $\text{[math]}$ . Since we already have all the necessary data, we will only use the formula to calculate it:

$\text{[math]}$

Thus, the compound interest would be $\text{[math]}$ after five years.

Calculate the final capital after six months, given an initial capital of $10,000 and a compound interest rate of 3.5% monthly

Solution

Note that we have the investment period, capital and interest rate

$\text{[math]}$
$\text{[math]}$
$\text{[math]}$

To solve this exercise we substitute directly into the formula to calculate the amount

$\text{[math]}$

Therefore, the capital is $12,292.55 after 6 months.

Find the nominal interest rate of an investment of $30,000 that after five years at an interest rate compounded quarterly becomes $50,000.

Solution

Note that we have the investment time, capital and amount

$\text{[math]}$
$\text{[math]}$
$\text{[math]}$
$\text{[math]}$

We calculate the investment period

$\text{[math]}$

Here we need to find the value $\text{[math]}$ . Since we already have all the necessary data, we will only use the formula to calculate it:

$\text{[math]}$

Substituting the values $\text{[math]}$ in the nominal interest rate formula we have

$\text{[math]}$

Therefore, the nominal interest rate is 10.35%.

In how much time will a capital of $30,000 produce an amount of $40,000 with a compound interest rate of 4%?

Solution

Note that we have amount, capital and interest rate

$\text{[math]}$
$\text{[math]}$
$\text{[math]}$

We substitute the data into the equation to calculate time

$\text{[math]}$

Therefore, the required time is 7.33 years.

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Agostina Babbo

Agostina Babbo is an English and Italian to Spanish translator and writer, specializing in product localization, legal content for tech, and team sports—particularly handball and e-sports. With a degree in Public Translation from the University of Buenos Aires and a Master's in Translation and New Technologies from ISTRAD/Universidad de Madrid, she brings both linguistic expertise and technical insight to her work.

Solved Exercises and Problems on Compound Interest

What Is Interest and How Is It Calculated?

Formulas for Calculating Compound Interest

Proposed Interest Calculation Exercises

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